or a similar form for the continuous case.
The MLE for , is any value of which maximizes
To maximize something we want to take the derivative and set it equal to zero. Often with MLE's it is easier to take the derivative of the .
The method of moments estimators come from matching the sample moments with the theoretical moments =function of the parameters.
If you have specific examples of ones you are having trouble with, you can post them and I can show you how to find these estimates. I think you are probably just as capable as anyone else of finding online proofs of these though.